1. Ke4 allows black to capture the knight, but once the white king captures black's h-pawn, there is no stopping white's promotion at h8.

If Black chooses instead to prevent 2.Kf3 by moving 1. ... Kg3, white will simply move 2.Kf5 (I originally posted 2.Kf4, but that was a typo error)-- still lined up to move 3.Kg4 no matter what black does. On his 4th move white will capture black's pawn, and there will be no stopping him from promoting his pawn at h8.

Nice endgame study! Couldn't see why 2.Kf4 was no good at first - had to step through the moves to see why it doesn't work.

Thanks for raising this issue, stepen_33.

2. Kf3 was the obvious choice, but it "looks like" 2.Kf4 should work just as well. BUT 2. ... Kf2!, 3. Kg4 Ke3, 4. Kxh4 Kf4 and now the white king is permanently pinned to the h-file, and the game is a draw.